A statistical mechanics of interacting biological species

Abstract

The system of differential equations proposed by V. Volterra, desoribing the variation in time of the populationsN _r of interacting species in a biological association, admits a Liouville's theorem (when logN _r are used as variables) and a universal integral of “motion”. Gibbs' microcanonical and canonical ensembles can then provide a thermodynamic description of the association in the large. The “temperature” measures in one number common to all species the mean-square deviations of theN _r from their average values. There are several equipartition theorems, susceptible of direct experimental test, a theorem on the flow of “heat” (the conserved quantity in an isolated association) between two weakly coupled associations at different temperatures, a Dulong-Petit law for the heat capacity, and an analog of the second law of thermodynamics expressing the tendency of an association to decline into an equilibrium state of maximal entropy. The analog of the Maxwell-Boltzmann law is a distribution of intrinsic abundance for each species which has been successfully used by ecologists for interpreting experimental data. A true thermodynamics develops upon introducing the idea of work done on an association through a variation of the variables (such as physical temperature) defining the physical and chemical environment. An ergodic theorem is suggested by the agreement of ensemble and time averages in the one case where the latter may be found explicitly.